Math for New Year's Eve!

Today is December 31st, also known as New Year's Eve, a time when we say good bye to the old year and hello to the next year.  Its a time of partying, of drinking, of eating herring or beans or other family tradition.  It also provides a wonderful set of activities we can integrate into the math classroom.

Yummy Math has a lovely activity focused on the Times Square celebration.  It provides a time line of the Times Square ball from 1907 to the present.  The activity asks students "What they wonder?" and "What questions they have?".

When I saw the data, my first thought was to graph it by diameter, weight, or number of bulbs used in the ball.  From a bit of research I've done, the pole is 141 feet and the ball is geared to drop in 60 seconds even back in 1907.

However, it is possible to research the number of people who go to watch the ball drop and create a ratio of number in the square vs the number of citizens in New York City at the same time.  For instance, I know there were 200,000 who turned out of a population of approximately 6 million people.  Since most of my students have not been out of Alaska, they have no idea what Times Square looks like.  I'll be honest, I am only familiar with the picture they show but have no idea what it looks like.

This is where Google Earth comes in handy.  It is possible to pull up the area where the Times Square ball is dropped so students can explore it using street view. They can also look at the over head view to see what the area looks like and get a better idea of the restrictions placed on the number of people who can view the ball dropping.

Its possible to find the numbers of people who visited Times Square over various years and find approximate populations for New York City at the same time to find a ratio.  It is possible to also determine the density for the area based on a bit of research.

Although viewing the dropping of the ball doesn't cost anything, it requires people start assembling in the area around the ball by 3 PM but if you want a prime view with dinner in a warm area, then people buy tickets for hotel activities.  This can easily be researched on line.  I did a quick search and discovered tickets are available for $999 to about $1400 per person depending on the hotel.

The costs of a party at a hotel provides possibilities for creating infographics, charts comparing what you get when you purchase a ticket to watch the ball from the warmth of a penthouse event.

Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

5 Ways to Engage Students

 One of the hardest things I have to deal with, is that of motivating students who see little use for school or for math.  Their parents and grandparents may never have finished high school, yet are doing well within the context of living in an isolated village.

Unfortunately about 52% of registered students are absent at any one time which can make it difficult for students to learn the material because they end up missing foundational pieces.  I believe this is why many of my incoming freshmen have significant gaps in their knowledge of mathematics.  Of the 48% who make it to school another 10 to 15 percent either sleep or do absolutely nothing because they either don't have the skills or have exhibited this behavior in previous grades.

One thing that is going to help students during the second semester is the iPads that were finally issued to my classroom.  I've been waiting for them to be distributed since August and I only got them at the beginning of December about 1.5 weeks before holidays.  Now I can post thing on Google Classroom so students can keep up with material when they are absent or are traveling.

So what are some ways that might help students become more motivated.  Motivated to come to class.  Motivated to keep up with work.  Motivated to even try.  Some of the suggestions I've read will not make a difference to motivating my students but some will.

1.  Relate the mathematics they are learning to the real world and to the world they live in.  Since my students live in a village accessible by air, trying to do problems with trains or automobiles leaving two different cities makes no sense but if I rewrite those problems using places they know and vehicles such as ATV's or Snow Machines, they can then relate to the problems.

I know one teacher who taught her students the words and pictures for items they'd likely encounter on a standardized test and equated the items to ones they understood.  For instance, many students fish using a piece of board and fishing line rather than a rod and reel so she taught them the rod and reel while relating it to their more traditional fishing methods.

2. Allow students to divide up in small groups of three to four people who will work on the problem assigned by the teacher.  The first group with the correct answer and the group with the highest number of points at the end of the period receive a prize. Another activity along this line is to provide a menu where students choose certain problems to do.  Students do well with choice so they can work problems they feel better able to do.

3. Use props to grab their attention or to illustrate formulas or concepts being taught.  Props can help students learn the material better.  If they are engaged, they are learning.

4. Assign problems with more than one answer.  Let the students know there is more than one answer and challenge them to find as many as they can.  Or chose a problem with more than one way of ding it and challenge students to find the multiple ways.

5.  Finally set up an environment which allows students to feel safe.  If they don't feel safe, they won't try and will set themselves up for failure but if they feel safe, they will try.  Its a matter of balance.

Let me know what you think, I'd love to hear.  Have a great weekend.

Writing Tiered Lesson Plans

  Now for information to write tiered lesson plans to go with the tiered assignments from yesterday.  It is something I need to really look at since I discovered too many of my students test at grades 4 and 5.

There are eight steps to look at each time the teacher writes a tiered lesson plan.  Understand that depending on the criteria used to divide students into groups may require the teacher to adjust groups as needed.

So look at these things every time you write a tiered lesson plan.

1.  Identify the grade level and subject for the lesson.

2.  Look at the standard you will be addressing in this lesson.  Don't wait to the end, do it now.

3. Identify the key concept or idea.  Ask yourself what is the big idea and what you want the students to know at the end of the lesson.

4. Decide if students have the necessary background knowledge or do you have to provide it?  Is scaffolding necessary?

5. Determine if content (what you want them to learn), process(how students make sense out of content) or product(outcome of the lesson).  It is recommended to only do one if you are just beginning.

6. Determine what type of grouping you should do.  One way is to divide students based upon pretest assessment.

7. Divide the students into groups.  It is recommended one use only three groups because if you go for more, it becomes more difficult.  The standard divisions could be into below level, on level, and above level.

8. Develop the assessment associated with the lesson which could be formative, summative, or both.

So now you know the basics for writing tiered lesson plans with tiered assignments.  Let me know what you think, I'd love to hear.

Creating Tiered Assignments

 Many of us work with students who see any math problem and automatically see it as "too hard".  They won't even try because they feel they can't do this.  One way around it, is to use tiered assignments because students have a choice and can choose those problems they think they can do.

Tiered assignments are a form of differentiated instructions that can be as simple as letting students choose problems based on their ability to having a much more complex structure.

Lets look at how to create tiered assignments for your classroom so you meet the needs of everyone, not just one group.  In order to create a tiered assignment there are some things you have to think about first.

The reasons for using tiered assignments include having students begin where they are, reinforces or extends material, allows students to have work which they do not perceive as too hard.  It promotes success.  

Before writing the assignment, you need to decide what part can be tiered?  Is it content? or process? or product?  Is this material being used to check readiness, interest, or test scores?  Then when you begin writing the actual assignment make sure it follows these suggested guidelines.

1. The task needs to cover the key concepts or generalization that is essential to the topic.

2. Use a variety of materials of differing levels of difficulty.

3. Adjust the task by complexity, number of steps, and independence.

4.  Let students know the grading criteria.

A good way to create the actual lesson is to answer these three questions.

1. What is the range of learning abilities?
2. What should students know, understand, and be able to do at the end of the lesson?
3. How will you hook students? What will they be able to do at the end of the lesson?
4.  Prepare three levels of the same assignment from easy to more challenging.

This process can be used for preparing lessons taught in the class and for daily assignments.  Tomorrow, we'll look at writing tiered lessons in detail to go with the tiered assignments.

Let me know what you think, I'd love to hear.  Have a great day.

Merry Christmas

The Math of Christmas Eve.

  I found it!  I found an article where someone looked at how many children,Santa has to visit on Christmas Eve so each and everyone gets a present.

The author explained where he got his numbers and what criteria he used for selecting numbers.  For instance he looked for the number of christian children aged fourteen and under.  he found there were 526,000,000 who celebrate Christmas on December 25th.

So if one assumes you have 24 time zones Santa has to travel through beginning late on the 24th so the gifts are there for the children in the morning, the author calculated Santa has to deliver 22 million presents each hour.   This breaks down to 365,000 children per minute or 6,100 children per second.

In addition, there are some wonderful graphs for students to practice reading.  One of the graphs breaks down the population according to time zones while the other looks at areas with the 22 million children in a time zone.  The two graphs compliment each other

Then towards the bottom, there is a short slide show discussing which country or countries are the most populous in each time zone, its population of children, and how long it should take Santa to deliver the presents.

Furthermore, at the bottom of the article, the author takes time to list issues which might effect the actual numbers such as not all Christians celebrate Christmas such as Jehovah's Witnesses and some non-Christians do celebrate it so that might change the numbers a bit.

There are also issues where certain countries cover several time zones which makes calculating things a bit tougher.  I really liked the article because the author really takes time to break it all down.  It is well written and worth using in class because its something you can assign to students to read and if you provide an accompanying worksheet with questions for students to answer.

Check it out.  Let me know what you think, I'd love to hear.  Have a great Christmas.

Alaskan Christmas

Found in the Anchorage Alaska airport by Security.  Warm-ups return after the New Year.

Thoughts

I have seen this guy entertaining travelers who are making their way through Sea-Tac airport.  I wonder how many have seen him or stopped to listen.  Sorry, its not a great shot but he was on the move and didn't stop.  He's a one man marching band.

Learned Something New

Yesterday was one of those days where our flight was delayed due to one of the more unusual issues.  Usually, we have planes run late due to weather, ice fogs, fog, or something similar but yesterday it was labeled as "maintenance".

I called the airline to discover they hadn't properly put their Navajo Caravans to bed the night before so when the temperature really dropped, the electronics got messed up.

They had to fix systems such as the auto pilot and other things in order to come pick us up.  I knew that if you don't plug your car in during super cold weather, the battery might loose power or the oil might get too thick but I never thought about electric circuits.

In the process of learning this, I realized I have the basis of a great lesson based on temperatures and the relation to the effectiveness of electronics.  I discovered that certain companies set electronics to work between certain temperatures such as Apple which sets their iPhones, etc to work well between 32 and 95 degrees F while Samsung sets their electronics to function best between -4 and 122 degrees F.

I figure I can have students research these tolerances for various electronics from different manufactures to create graphs based on those tolerances.  From the graph, they can provide a written paragraph explaining which manufacturer they recommend.  This is very appropriate for us since temperatures regularly drop to the -20 F range and colder.

In addition, I just discovered that the effective range depends on if it commercial, industrial, or military based with military based having the best spread for temperature range.  I'm also sure with a bit more research they could find graphs showing more of a degree by degree connection with effective operation of the electronics. 

Furthermore, since most of these items operate off of battery power, the project could include a section on the temperature range of effective use for batteries.  In cold weather, we have to wrap the car batteries in a battery blanket so it stays warm over night and the battery does not loose it charge.  For up here, we have to get the heavy duty batteries designed for the low temperatures. 

So many possibilities and my students can actually relate to this because they have to start a ATV or snow machine in cold weather and its a challenge.  Many throw blankets over their machines while others use a hair dryer to warm everything up.

Let me know what you think, I'd love to hear.  Have a great day.

Interesting Training

  Yesterday, we had a very interesting training after school.  Just a bit of background first.  The super decided to transform our school from one principal and one assistant to two principals.  He divided the school into elementary with one and middle/high with the other.

So yesterday, the soon to be middle school/high school principal ran a training on what he wants in his lesson plans.

He is the first person in all the time I've taught at this school to do that. He even modeled it which was great.  He wanted to do this so we'd know what to expect when we got back.  So here it is:

1.  Write a simple objective.  It doesn't have to be anything fancy but it has to be able to be assessed so you know if you met the objective. 

2.  This is where you teach the main lesson.  No student should be talking and it shouldn't be longer than say 11 to 15 min at most.

3. This is where you check for understanding through discussion or centers.  You are expected to wander around checking and making sure they are on task.  If they are engaged, let them talk longer.

4.  This is where you have students talk in a think, pair, share, or give each person in a pair a different problem that they do and then teach the other one.  Keep them busy.

5. Only after you know the student knows how to do the work, do you give them the assignment which will be graded. 

6.  The final step is providing closure.  Provide the answer of when and where this will be used in the village or in real life.  Unfortunately, for math it gets a bit harder because we only have a once a week newspaper with few graphs and few sports results.  I know there are two stop signs but no children at play or yield signs.

He also said that we needed to put the lesson plan in the classroom where he can get to it easily and check to see if we are doing Tuesday's lesson on Tuesday rather than Thursday but that tends to happen out here a lot. 

The best news of all is that I finally got my iPads so now I can set up my google classroom, and other things.  The bad thing is they haven't put all the apps I've requested so I don't know if they will do that over holidays.

I'm off, have a great day.

Christmas Program

Today is the Christmas program, one day before holidays.  Picture 300 students grades pre-K to 12th all going on to entertain parents but first the head start children file on, look cute, and leave.  It will take a couple of hours to complete.  I've spent the past two days, offering students a chance to pass so they are eligible to play basketball next semester.  Should be back to normal tomorrow.

The Mother of Cryptology

The modern mother of cryptology is a woman whose achievements were lost in time due to others claiming responsibility for what she did. 

 Elizebeth Smith Friedman is a woman whose name is not one that immediately pops to mind when discussing cryptography or mathematics.  In reality, her husband is much more well known since he helped found the NSA.

Elizebeth was born in 1892, the youngest of 9 children.  Although she received her degree in English lit with strong studies in Latin, Greek, and German, she ended up applying for a library job that resulted in being hired at a private think tank in the late nineteen teens.

At this point in the, this facility had the only national cryptologic laboratory devoted to trying to prove Sir Frances Bacon actually wrote all of Shakespeare's work.  It was here that she met and married William Friedman, her husband and partner.  After a few years, they moved to Washington, D.C. to work for the government.

Although a poet and not a mathematician, she taught herself how to decode secret messages without knowing the key.  Her mind was incredible to crack codes, finding keys, and creating methods that changed the world of cryptography. Furthermore, she helped invent cryptology or  the modern science of secret writing.

One of her first jobs for the Navy and Coast Guard had her breaking codes used by rum runners or those who illegally transported alcohol and other goods during prohibition.  She helped capture criminals and testified at numerous trials which resulted in convictions because she easily explained how she cracked the codes to juries. During this time, she cracked over 20,000 messages whether simple code, transposition, or something more complex.

In 1937, she helped the Canadian government convict an opium dealer by cracking the code based on Mandarin Chinese without knowing the language. Just a few years later, she and her team of code breakers began intercepting messages that were quite similar to the prohibition type messages but were sent by Nazi spies.

Just before World War II, she transferred to the Coordination of Information where she
focused on cracking certain Enigma codes, specifically those based in South America, so messages could be intercepted.  She shared the codes with the FBI, giving them the ability to intercept and translate messages resulting in the South American spy network being shut down.  J. Edger Hoover claimed the FBI did all this on their own and ensured no one knew she had anything to do with it by creating a propaganda film.

She also discovered the letters written by Velvalee Dickinson contained coded information about the moves of ships at Pearl Harbor.  Her work was responsible for Velvalee's conviction in 1944 as a Japanese spy.  Five of her letters were sent to Elizebeth because the letters with their talk about dolls in her doll collection seemed strange.  Velvalee did own a doll shop in New York City where some of the dolls sold for as much as $750 each but upon investigation, it was noted that she had fallen into debt upon her husband's death.  Investigators discovered her ties to the Japanese consulate and the Japanese American Society which was enough to arrest and try her.

Although she had no direct link to beginning the National Security Agency,  she helped create the science underlying their work. It was her husband who helped found it with the Army code breaking unit he founded in the 1930's that was later absorbed into the NSA.

If you are interested in her story, check out the book "The Woman Who Smashed Codes" by Jason Fagone.  Let me know what you think, I'd love to hear.  Have a great day.

Knitting Magazine

  I collect knitting books and magazines for the day I have more time to enjoy myself.  I picked up a magazine that was not the usual one filled with capes, sweaters, and tops.  It was filled with wonderfully mathematically based knits.

I've written about this topic before but all the pieces were spread out all over the internet but these projects are all in the "Creative Knitting special issues" from April 2017 that is filled with products using 1, 2, or 3 skeins.  I had a blast reading it mathematically.

It started off with something alledan infinity cowl.  It might be because it offered multiple ways to wear the cowl so each time it was put on but that wasn't the one that caught my attention.  It was the Ridged Moebius cowl, one could knit that is large enough to wear over the shoulders.  If you didn't like that one, you could always make the Roman Stripe Moebius. 

A few pages later in the issue, I stumbled across three different sized Modern Cubist Baskets which are either proper cubes or a rectangular prism with handles that one can place knitting in.  The bottom and sides are knitted separately knitted before being attached.

Immediately following this project, I found instructions for making nautical coasters in circles, hexagons, and squares.  The authors even offer a square within a square.  Of course, if coasters are not the thing, then try potholders which are square in shape but have fascinating patterns such as a pot holder made of blocks created by the pattern.

Next come the instructions for making felted bags in a squarish shape.  They are first knitted, then felted with handles that gather the mouth of the bag closed. This is followed by headbands that are nothing more than long rectangular strips of knitted material attached to form a circle.

Talk about being in heaven, Moebius strips, cubes, rectangular prisms, squares, hexagons, circles, and more.  So many mathematical concepts rendered in knits.  Furthermore, knitting is filled with patterns, patterns, and more patterns created to take one long piece of yarn and turn it into something so much more.

While I am out at Christmas time, I think I might stop and pick up some needles and yard to make these things over the next few months.  No I'm not a great knitter, I'm able to follow patterns well enough to make a pair of socks but I'm not quite up to making lace. 

There are enough patterns in this book and on the internet to have for a after school mathematical knitting club.

Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

School Based Business or Store.

  The business teacher at school is new but brings with her a plan.  She went to Donors choice to get the funding for equipment to cook hamburgers and fries so one of her classes could learn to run a business.

The idea is for all the students in that class to obtain their food handlers licenses to prepare them for jobs after school.

In addition, they have to figure out much meat, buns, fries, sliced cheese, and condiments to order for an event.  This means, they are using math when they fill out the purchase order because everything has to be ordered in from Anchorage and shipping can be quite expensive because it is air freighted.

Another thing they end up doing mathematically is calculating the amount they can sell the hamburgers and fries for so they make a profit and so people will buy their product.  Its a fine line sometimes because if the price is too high, they may not sell everything and loose money.

Out here a good hamburger can go for $25.00 with fries costing $5.00 for some but the school isn't charging that amount because they feel they can still make a decent profit without going that high.

Furthermore, we have a concession stand to sell things like soda, chips, and candy to  people at various activities, sports games, etc.  The senior class has it during the first semester while the juniors take over for the second semester.  Students usually sit down with the local Span Alaska catalogue to buy things by the case.

Span Alaska is a company who has been selling case lots and individual cans via catalogue in Alaska since 1972.  The price they quote usually includes the shipping already tacked on but if it requires special handling or is frozen, shipping is added on. 

Students order soda, gatorade, candy, chips, and other goodies to sell in the concession stand.  Since the price includes shipping, they know what the total cost is for the purchases so its easy to determine the cost they paid for each item.  Based on this, they can figure out how much of a mark-up for the selling price but they also have to keep track of how much the item sells at the local stores.  If they are too much out of line with the rest of town, many people will pop over to the store to buy things.

The idea is that students use this as a fund raising activity for their last year in preparation for graduation.  If the class has earned enough money over the four years, they might take a trip to Hawaii or to California.  Its a great experience because they have to determine how much it will cost to fly to a certain destination, rent cars, rent hotels, include spending money, etc.  In other words they have to create a budget and a goal.

Although both projects require a lot of work, it gives students a great experience in real life application of math, especially math geared for running a business. Let me know what you think, I'd love to hear.  Have a great weekend.

The 2018 Cost of the 12 Days of Christmas

I grew up listening to the 12 days of Christmas, both the traditional version and the Hawaiian version.  I'll admit that for the longest time, I thought it was 3 French Horns because I'd never heard of a French Hen.

It was only after I started to read, I discovered how far off I was.  Around Christmas time, it is possible to find the current cost of giving the 12 days worth of gifts.

This year, I managed to find a wonderful article in Forbes which included the increase or decrease for each item. According to the article, published in November, it will cost $39,094.93 to buy all these gifts.  This is an increase of $450 over last year which comes out to a 1.2% increase.

The list, including increases is:

•  1 Partridge in a Pear Tree: $220.13 (+.1%)
• 2 Turtledoves: $375.00 (no change)
• 3 French Hens: $181.50 (no change)
• 4 Calling Birds: $599.96 (no change)
• 5 Gold Rings: $750.00 (-9.1%)
• 6 Geese-a-Laying: $390.00 (+8.3%)
• 7 Swans-a-Swimming: $13,125.00 (no change)
• 8 Maids-a-Milking: $58.00 (no change)
• 9 Ladies Dancing: $7,552.84 (no change)
• 10 Lords-a-Leaping: $10,000 (+3.0%)
• 11 Pipers Piping: $2,808.40 (+3.5%)
• 12 Drummers Drumming: $3,038.10 (+3.5%)

Much of this information comes from the PNC Christmas Price Index which for the past 35 years has calculated total cost complete with increases and decreases.  The method used to do the calculations is almost the same process as the government uses to calculate the Consumer Price Index.

Their list is done for each item with a infographic like graphic for each item along with the percent change.  Furthermore at the bottom of the webpage is a graph showing the increase of the cost, year by year, beginning in 1984.  This is a great graphic to read and interpret.

In addition, at the bottom of the page is a 12 page activity guide with everything needed  to teach lessons on this topic.  The activity includes reading data, estimating, graphing, and requires students to explain their rationale for certain things.

I know what I'll be doing next week on the last couple of days before holidays.  Let me know what you think, I'd love to hear.  Have a great day.

Why Use Exit Tickets?

We see recommendations to use exit tickets regularly in class but one often wonders why use them in the math class.  Why even use them at all because its one more thing to use in class because its just one more thing to keep track of in class?

Exit tickets are like thermometers where the teacher checks overall student understanding.  They can be used to check for student understanding on a topic so the teacher knows if they need reteaching or if they are ready to move on.

In addition, the exit ticket provides information on if the whole concept needs reteaching or if a small point needs clarification such as if you multiply or divide by a negative, the sign changes.

Furthermore, exit tickets help students understand that the material is important and they are accountable for learning it.  It helps them synthesize the material, helps them move it from short term to long term memory because they are accessing it.

Another plus for using exit tickets is that students have to learn to communicate in writing.  If the teacher checks answers and has questions, they can clarify points to understand student thinking better.

If you do not currently use exit tickets, it is recommended you start slow.  Perhaps use it once a week and only on one or two topics.  It is suggested teachers do not grade exit tickets because its an assessment tool designed to provide data for instruction.  It lets the teacher focus on students who still don't quite have it with a bit extra instruction while provided more advanced problems for those who "have it".

To create an effective exit ticket does not take much.  Just follow a few simple rules:

1.  It is linked to the objective of the lesson.
2.  Focus on one skill or concept taught that day.
3.  Questions may be multiple choice, short answer, or require a couple of sentences.
4.  Exit tickets should have no more than 5 questions but fewer are usually better.
5.  Students should be able to finish the ticket in a few minutes.

The questions should not require a simple yes or no answer because that give no information.  Exit tickets should have questions that assess understanding, allow student to demonstrate the concept through work, or application of the concept.  You might create a problem based on the day's topic that could show up on a test redone, discuss how the topic could be used in real life, rate your understanding of the topic based on a 1 to 10 scale, or write a short paragraph on the day's lesson.

In addition, it is quite easy to set up digital exit tickets using google forms, or other app so you don't have tons of paper to keep track of.  More on this topic in a while.  Let me know what you think, I'd love to hear.  Have a great day.

Multiple Choice Questions - Other Uses.

When most of us hear the term "Multiple Choice" in connection to math, we automatically joke "Multiple Guess".  I remember taking a few where I just plugged the answers back into the problem to find the correct choice, or I used the elimination method to narrow the choices until I got it down to the most probable one.

Many multiple choice questions are set up so the most probable incorrect answers are included so if someone makes a mistake, they'll find it in the choices.  More often than not, you aren't sure if they know the material, got tired, or didn't care.

The other day, I read a blog on how to use multiple choice questions in ways that may be more effective in class. There are ways to use it other than as a major test.  I liked what the author said so I'll share it with you.

I'd like to thank Pear Deck for these suggestions  which are easily integrated into math.

1.  Use multiple choice questions to poll students on a topic or get feedback on something.  You might ask them if they'd prefer to play kahoot or jeopardy as a way of reviewing the material in preparation for a test.  You might also ask which step is next in a problem by listing several steps to  see if they understand the process. 

2.  Multiple choice can be used to check for misconceptions and understanding in topics such as
      GCF & LCM
      Equivalent Fractions
      Order of Operations
      Binomial Multiplication
      Combining like terms
      Solving problems
The use of whiteboards either virtual or real is great for this because you post the question, students write their answer on the board and with a quick glance, you see who understands it and who needs a bit more work.

3. Use multiple choice as an exit ticket.  Ask how they feel about the material studied that day using emoji's and multiple choice.  Faces can range from great to crying so students can choose one to show how they feel about their understanding.

These are just a few suggestions for using multiple choice to learn more about student understanding outside of a testing situation.  I think they are cool and add to my teaching toolbox.  Let me know what you think, I'd love to hear.  Have a great day.

Compare and Contrast in Math

Compare and contrast is something often used in Social Studies, English, or other subject but it is not something used as often in math, not because its too hard but because most math teachers have not been trained to use it.

The closest thing we have is the Vann Diagram but its not exactly a compare and contrast exercise.  So with a bit of thinking and working, it can be used in the math class but not for every topic.

First its important to know that when compare and contrast is used, it can strengthen the student's memory, help develop higher order thinking skills, improve comprehension, precision and helps build good work habits.

Lets look at some ways to use compare and contrast in math.

1.  Compare and contrast inequality signs.  Many students have difficulty in distinguishing between less than, more than, greater than or equal and less than or equal.  Requiring them to fill out a chart comparing similarities and contrasting or finding differences can help them put into words their understanding.

2.  Compare and contrast place values such as tenths and tens, thousands and thousandths, so students learn to tell the difference between decimal values and whole number values.  They are very similar but with a small difference.  Some of my students arrive in 9th grade not knowing their place values which hurts them at times.

3. Long division and synthetic division in Algebra.  Although they appear different, the methods share quite a few similarities.  Using a compare and contrast will show students their similarities and differences.  In addition, they can see that it doesn't matter which method used, the answer is the same.

4. Compare and contrast distributive property with binomial multiplication.  It doesn't really matter what method you use to complete binomial multiplication but it still involves distributing terms, just like we teach when using distributive property.

5.  Compare and contrast Greatest Common Factor with Lowest Common Multiple. I can tell you, my students are having so many problems with this topic so completing a compare and contrast in the hopes they can learn to distinguish between the two.

6.  Compare and Contrasting congruent with similar triangles since both share similar methods of proof. 

7.  Compare and contrast bisectors with medians and altitudes because students often have trouble remembering the differences among them.

8. Compare and contrast area with surface area.  I realize one is 2 dimensional while the other is 3 dimensional but they do share similarities.

Any two math topics which have similarities and differences can be used in a compare and contrast exercise.  Let me know what you think, I'd love to hear.  Have a great day.

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Warm-up

December 7, 1941

Today is the day to remember an event that happened 72 years ago.  Japan bombed Pearl Harbor, killing both military and civilians while trying to cripple the American fleet.

Its sometimes difficult to find activities designed specifically for the math classroom.  Most of the ones I've found are for history or English but that has not stopped me at all.

This particular day has lots of possible activities which you can provide the data for, or you can have them find the data for the project.

1.  Its easy to find the number of people killed in each branch and civilians on December 7, 1941.  This information can be turned into a chart to show the information visually or it could be turned into an infographic with the graph.

2.  Another graph could be done showing those from each branch and civilians who were wounded during the attack.  You can find the information from the Honolulu paper showing how many and the ages of civilians who were killed by bombs or bullets from the attack.  Again, its easy to create a graph showing age distribution.

3.  Number and types of ships that were damaged or sunk. The Japanese wanted to sink aircraft carriers but there were none in port that day.  Instead they got battleships, destroyers, etc. This exercise could also include the number of planes destroyed or it could be done seperately.

4.  Find out where the mini-subs departed from the mother subs, when they left and when they arrived at pearl harbor.  This is enough information to calculate their average speed or you might find the average speed to determine how long it took them to travel the distance.  Furthermore, its possible to calculate the volume of the two man mini-subs.

5.  Another activity dealing with distance is to find out where the Japanese planes took off from, find the distance, and how long they took to get to Pearl Harbor to determine the average rate of speed. 

6.  Find out how many total planes the Japanese sent to bomb Pearl Harbor and the number of each type of plane to create a graph.

7.  Students can also create an infographic containing all sorts of numerical information on the attack on Pearl Harbor. 

Let me know what you think, I'd love to hear.  Have a great day.

Modeling Viral Fake News.

It used to be, people would open a newspaper, watch television, or listen to the radio to discover what was happening in the world.  If you were lucky, you might see it happen live on television such as during an earthquake when the world started shaking and you saw the newscasters dive under their tables for safety.

Now, news is more instantaneous with the use of social media.  You can see things as they happen because people can record and instantly post on any one of the numerous social media sites.  You can also see people add 2 + 2 to get 5 instead of 4 and this is when we see fake news going viral.  There are people out there who have worked out the math on why fake news goes viral.

First of all, the way social media is set up, just about anything can go viral due to the amount of information out there and the inability of people to fully evaluate every piece they see.  Most of these pieces of "news" include a video, picture, link, phrase, or other form of online information and the "reader" has to sort through so much.  Second, there is the amount of time people spend looking at any item so if it doesn't capture their attention, they'll skip it and share something that does.  Finally, the way social media is set up, it encourages  indiscriminate sharing.

Now as far as the math goes, social media uses agent-based models because individuals are the ones who share things.  This is the same model used to determine how disease spreads through communities.  If you were to visualize it, you'd see dots representing the individual and arrows pointing from the individual to others as they shared the disease or fake news.

Mathematicians have had to modify it a bit because the original model is for one disease not thousands of pieces of information shooting across the internet each day so they've had to include the probability of a person making a new piece of information or looking through things they've gotten before sending it on.  In other words, they look at the most likely attention span of the individual for sorting through all the messages before sending one on.

People have speculated that super connected people on social media are more likely to cause something to go viral but one scientist looked at that speculation and concluded it is not true.  She stated, most of these super connected people do not have time to go through all the material they receive and they certainly don't have time to send everything they might want to.

What is more likely is that groups or clusters of people are more likely to socially share, creating a social reinforcement of material because every time you see it, it becomes more believable.  If its believable, it has to be true right?  This is one reason certain pieces of fake news go viral.

Scientists and mathematicians are still working out the intricacies of this topic but they are getting closer.  Let me know what you think, I'd love to hear.  Have a great day.

Math and The Stock Market.

Now that you know why the stock market used fractions, its time to look at the math involved in the stock market and there is quite a bit.

First off, anyone who has a portfolio with a broker or of their own, knows the portfolio is divided into a pie chart showing how much is in stocks, bonds, cash, and mutual funds.  If your portfolio does not have that much diversity, it could show how many shares of each stock you've invested in.

Second, it is possible to calculate the return or money you are making off your portfolio by using this equation.  (Current Value/Starting Value - 1) * 100 tell you how much you made or lost for the entire portfolio.  This is important because so many people are interesting in building a portfolio which will support them for their lives, throughout their retirement.

Third, people calculate the stock return so they know if a stock is performing poorly or well so they know if they want to keep it or sell it off.  The formula is similar to the last one but this time its
(Last price quoted/price paid - 1) * 100.  This formula can be applied to each stock to determine its return so you know if you want to keep it.

Here are other equations associated with the stock market that investors need to know.

1. Earnings per share is a part of a company's profit set aside for common shares which is an indicator of how profitable a company is.  The formula (Net Income - Preferred Dividends)/Total Outstanding Shares.

2. Return on Equity also known as a rate of return on net assets often is associated with the company showing its financial performance.  It is found by (Net Profit/Shareholders Equity * 100 ) which give a percentage. 

Now if you check this site out  there are lessons on Risk and Return, Return on Investment, Investing Options, Stocks and Stock Market, Stock Investment Analysis, and Bonds.  The Stock Market lessons are quite interesting because there are lessons on investing in stocks, stock market table, stock market simulation, buying shares and determining how many shares one can afford, percent price change in shares, and forms to fill out for buying and selling stocks.

In the Stock Investment Analysis, there are exercises for price earnings ratios for stocks, and common stock valuations while the Bonds section has information on buying and investing in corporate bonds.  Information that few people really understand.  Even I have someone who does all that for me because I don't know how to go about it.

The stock market simulation uses most of the information provided to give students an experience in investing in the stock market.  This activity could take as little as a week but its recommended the simulation be run the full year to give students a better understanding of how it works.  Throughout the designated time period, all students invest a specific amount in 5 stocks which they will follow over a certain time period.  they may buy and sell but they must pay a 3% commission each time they buy or sell a stock.

They have to fill out transaction forms every time they buy or sell stock, monitor the portfolio and if you want at this point, you could have them calculate the overall worth increase or decrease of its value each week.  This process continues until the end of the period at which point they sell off all the stocks to determine how much money they made or lost overall.

Its a nice basic simulation that you as teacher could add to so students are doing a bit more mathematics.  Its up to you.

Let me know what you think, I'd love to hear.  Have a great day.

The Stock Market Uses Decimals Too!

  When I was in school, one of the common examples of needing to know fractions included the stock market because they would show everything in eights.  For instance Apple might go up 1 3/8 points while General Electric might drop 1/8 of a point.

Just a bit of background on the stock market and why it used fractions rather than decimals originally.  The New York Stock Exchange was formed back in 1792 due to the Buttonwood Agreement.

 24 leading bankers, brokers, and merchants agreed to create a central clearing house for trading stocks and securities.  These men modeled their exchange on the one in Spain after checking out others in Europe because the United States currency had been based on the Spanish Real.

The Spanish silver dollar or Real was divided into two, four, or eight parts which is where the term "pieces of eight" came from because they could count them on their fingers. The Spaniards did not count using their thumbs like the English did.  So when the stock market began, they based the smallest increase on 1/8 of a dollar or 12.5 cents. In other words, if one stock dropped 1/8th you'd only loose 12.5 cents but what if you had 100,000 shares of the stock you'd loose $0.125 x 100,000 or $12,500 which adds up. 

This spread could cause people to gain lots of money when dealing in millions so they dropped the increase or spread to 1/16 or 6.25 cents.  Along the way, some stocks used a spread of 1/32 or 1/64 to keep it much smaller.

In 1997, the Common Cents Stock Pricing Act was passed by congress to make it easier for people to understand the pricing system since more and more people were investing in it.  Although, stocks began changing over in August 2000, it took till February 2001 for all 3025 companies listed on the NYSE to convert from fractions to decimals.  This change encompassed about 280 million shares.

Two of the things this did was to:
1. Investors could save over $1billion or more each year.
2. Investors could save on the cost of commission since commission was often based on the price of the stock.
3.  The United States is more compatible with other stock markets who have been using decimals for years.
4.  The number of transactions handled by the NYSE is able to double using decimals instead of fractions.

This was just a look to see why the stock market used fractions for most of its life and why it switched to decimals.

Let me know what you think, I'd love to hear.  Have a great day.

Moving From LCM to Adding Fractions Using Legos.

 As you know, I've been playing with Legos to figure out how to use them for something other than basic fractions in lower elementary.

I already figured out how to use them to help find Lowest Common Multiples or LCM so its really not a huge step to seeing why its important to have equivalent fractions when adding.

So I used the same number of Legos for the denominator and used a second set of Legos in a different color to represent the numerator. 

As you can see in the photo above, you see the 1/4 and 1/6 quite easily.  Then using the same method I added another 1/4 and 1/4 to get 3/12 which is the correct equivalent fraction.

For 1/6, it required a second 1/6 to get a total of 2/12.  This allows students to easily see why you multiply the numerator and denominator by the same number so you end up with equivalent fractions.

The same process could be used to show subtraction as well as addition so students can see it.

If you have enough Legos, it is possible to have larger denominators such as 15, 28, or 34 and allow them to see the process works as well on larger denominators as for smaller ones.

Let me know what you think.  I'd love to hear.  My next project is multiplication of fractions and whole numbers using Legos.  As soon as I have that done, I'll share it.

Warm-up

Warm-up